Conformally Invariant Variational Problems

Abstract

Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, $Q$-curvature, Yang-Mills fields, etc... We shall be concerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs arising from conformally invariant two dimensional variational problems (e.g. harmonic maps, prescribed mean curvature surfaces, Willmore and Constrained conformal surfaces, isothermic surfaces). The present manuscript are lecture notes of courses given by the author at several places including UBC Vancouver, SNS Pisa, IHP Paris, ICTP Trieste.

References in corpus (2)

• Global regularity of wave maps II. Small energy in two dimensions
• Sub-criticality of Schroedinger Systems with Antisymmetric Potentials

Cited by in corpus (11)

• Immersed Spheres of Finite Total Curvature into Manifolds
• Energy Quantization of Willmore surfaces at the boundary of the Moduli Space
• A Resolution of the Poisson Problem for Elastic Plates
• Higher Regularity of Weak Limits of Willmore Immersions I
• Noether Theorems for Lagrangians involving fractional Laplacians
• Existence of Min-Max Free Boundary Disks Releasing the Width of a Manifold
• Regularity of weak solutions to higher order elliptic systems in critical dimensions
• Elastic bending energy: a variational approach
• $L^p$ regularity theory for even order elliptic systems with antisymmetric first order potentials
• The Willmore conjecture
• The Lamm-Riviere system I: $L^p$ regularity theory